Results 11  20
of
37
Notions of computability at higher types II
 In preparation
, 2001
"... ntroduce some simple general theory to allow us to talk about notions of highertype computable functional. The following definitions (with minor variations) appear frequently in the literature. Definition 1.1 (Weak partial type structures) A weak partial type structure, or weak PTS A [over a set X ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
ntroduce some simple general theory to allow us to talk about notions of highertype computable functional. The following definitions (with minor variations) appear frequently in the literature. Definition 1.1 (Weak partial type structures) A weak partial type structure, or weak PTS A [over a set X], consists of the following data: . for each type #, a set A # of elements of type # [equipped with a canonical bijection A 0 # = X], . for each #, # , a partial application function ## : A ### A # # A # . We usually omit type subscripts from application operations, and often write x y simply as xy. By convention, w
On the computational complexity of Longley's H functional
 Presented at Second International Workshop on Implicit Computational Complexity, UC/Santa Barbara
, 2000
"... Longley [Lon98b] discovered a functional H that, when added to PCF, yields a language that computes exactly SR, the sequentially realizable functionals of van Oosten [vO99]. We show that if P ̸ = NP, then the computational complexity of H (and of similar SRfunctionals) is inherently infeasible. The ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Longley [Lon98b] discovered a functional H that, when added to PCF, yields a language that computes exactly SR, the sequentially realizable functionals of van Oosten [vO99]. We show that if P ̸ = NP, then the computational complexity of H (and of similar SRfunctionals) is inherently infeasible. The sequentially realizable functionals (denoted SR) is a class of “sequentially computable ” highertype functionals. This class, which includes the PCFcomputable functionals along with elements that fail to be Scottcontinuous1, has quite strong and natural mathematical properties [Lon98a, Lon98b, Lon99,
Linear Programming Languages
"... Abstract. We formalize SℓPCF, namely a programming language which is able to represent linear function between coherence spaces. We give an interpretation of SℓPCF into the model of linear coherence spaces and we show that such semantics is fully abstract with respect to it. SℓPCF is not syntactical ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We formalize SℓPCF, namely a programming language which is able to represent linear function between coherence spaces. We give an interpretation of SℓPCF into the model of linear coherence spaces and we show that such semantics is fully abstract with respect to it. SℓPCF is not syntactically linear, namely its programs can contain the same variable more than once. Last, we address the universality problem. 1
Matching typed and untyped realizability (Extended abstract)
"... Realizability interpretations of logics are given by saying what it means for computational objects of some kind to realize logical formulae. The computational objects in question might be drawn from an untyped universe of computation, such as a partial combinatory algebra, or they might be typed ob ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Realizability interpretations of logics are given by saying what it means for computational objects of some kind to realize logical formulae. The computational objects in question might be drawn from an untyped universe of computation, such as a partial combinatory algebra, or they might be typed objects such as terms of a PCFstyle programming language. In some instances, one can show that a particular untyped realizability interpretation matches a particular typed one, in the sense that they give the same set of realizable formulae. In this case, we have a very good fit indeed between the typed language and the untyped realizability model—we refer to this condition as (constructive) logical full abstraction. We give some examples of this situation for a variety of extensions of PCF. Of particular interest are some models that are logically fully abstract for typed languages including nonfunctional features. Our results establish connections between what is computable in various programming languages, and what is true inside various realizability toposes. We consider some examples of logical formulae to illustrate these ideas, in particular their application to exact realnumber computability. The present article summarizes the material I presented at the Domains IV workshop, plus a few subsequent developments; it is really an extended abstract for a projected journal paper. No proofs are included in the present version. 0
The elimination of nesting in SPCF
 In Proceedings of TLCA ’05, number 3461 in LNCS
, 2005
"... Abstract. We use a fully abstract denotational model to show that nested function calls and recursive definitions can be eliminated from SPCF (a typed functional language with simple nonlocal control operators) without losing expressiveness. We describe — via simple typing rules — an affine fragmen ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We use a fully abstract denotational model to show that nested function calls and recursive definitions can be eliminated from SPCF (a typed functional language with simple nonlocal control operators) without losing expressiveness. We describe — via simple typing rules — an affine fragment of SPCF in which function nesting and recursion (other than iteration) are not permitted. We prove that this affine fragment is fully expressive in the sense that every term of SPCF is observationally equivalent to an affine term. Our proof is based on the observation of Longley — already used to prove universality and full abstraction results for models of SPCF — that every type of SPCF is a retract of a firstorder type. We describe retractions of this kind which are definable in the affine fragment. This allows us to transform an arbitrary SPCF term into an affine one by mapping it to a firstorder term, obtaining an (affine) normal form, and then projecting back to the original type. In the case of finitary SPCF, the retraction is based on a simple induction, which yields bounds for the size of the resulting term. In the infinitary case, it is based on an analysis of the relationship between SPCF definable functions and strategies for computing them sequentially. 1
Inductive Definition and Domain Theoretic Properties of Fully Abstract Models for PCF and PCF+
 LOGICAL METHODS IN COMPUTER SCIENCE 3(3:7), 1–50 (2007)
, 2007
"... A construction of fully abstract typed models for PCF and PCF+ (i.e., PCF+ “parallel conditional function”), respectively, is presented. It is based on general notions of sequential computational strategies and wittingly consistent nondeterministic strategies introduced by the author in the sevent ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
A construction of fully abstract typed models for PCF and PCF+ (i.e., PCF+ “parallel conditional function”), respectively, is presented. It is based on general notions of sequential computational strategies and wittingly consistent nondeterministic strategies introduced by the author in the seventies. Although these notions of strategies are old, the definition of the fully abstract models is new, in that it is given levelbylevel in the finite type hierarchy. To prove full abstraction and nondcpo domain theoretic properties of these models, a theory of computational strategies is developed. This is also an alternative and, in a sense, an analogue to the later game strategy semantics approaches of Abramsky, Jagadeesan, and Malacaria; Hyland and Ong; and Nickau. In both cases of PCF and PCF+ there are definable universal (surjective) functionals from numerical functions to any given type, respectively, which also makes each of these models unique up to isomorphism. Although such models are nonomegacomplete and therefore not continuous in the traditional terminology, they are also proved to be sequentially complete (a weakened form of omegacompleteness), “naturally” continuous (with respect to existing directed “pointwise”, or “natural” lubs) and also “naturally” omegaalgebraic and “naturally” bounded complete—appropriate generalisation of the ordinary notions of domain theory to the case of nondcpos.
A Proof System for Correct Program Development
, 2000
"... realworld applications (e.g. [EHM + 99, Buh95]). Moreover, aspects of ML such as strong typing and the exceptions system have significantly influenced the design of languages such as Java [GJS96], and it seems likely that future systems languages will incorporate many of these features [Mac00]. ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
realworld applications (e.g. [EHM + 99, Buh95]). Moreover, aspects of ML such as strong typing and the exceptions system have significantly influenced the design of languages such as Java [GJS96], and it seems likely that future systems languages will incorporate many of these features [Mac00]. Regarding the second requirement, even before the definition of ML had fully taken shape, the LCF system [GMW78] provided a program logic for a rather restricted fragment of the language. Subsequent research has sought to build on the definition in order to support formal reasoning about programs. Most notably, the Extended ML project [KST97] resulted in a formal language for specifying program properties, but the complexity of this language prohibited the development of useful proof rules. A di#erent approach has been pursued by Elsa Gunter et al [GV94], who have formalized the definition of ML within the HOL theorem prover; this has proved useful for metatheo
Sequentiality in Bounded Biorders
 FUNDAMENTA INFORMATICAE
, 2005
"... We study a notion of bounded stable biorder, showing that the monotone and stable functions on such biorders are sequential. We construct bounded biorder models of a range of sequential, higherorder functional calculi, including unary PCF, (typed and untyped) callbyvalue and lazy λcalculi, and ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We study a notion of bounded stable biorder, showing that the monotone and stable functions on such biorders are sequential. We construct bounded biorder models of a range of sequential, higherorder functional calculi, including unary PCF, (typed and untyped) callbyvalue and lazy λcalculi, and nondeterministic SPCF. We prove universality and full abstraction results for these models by reduction to the case of unary PCF, for which we give a simple new argument to show that any orderextensional and sequential model is universal.